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          软考-算法思想1
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        <h4 id="算法思想的概览"><a href="#算法思想的概览" class="headerlink" title="算法思想的概览"></a>算法思想的概览</h4><p><img src="http://localhost:4000/2021/03/03/%E8%BD%AF%E8%80%83-%E7%AE%97%E6%B3%95%E6%80%9D%E6%83%B31/001.png" alt></p>
<span id="more"></span>
<h4 id="分治思想"><a href="#分治思想" class="headerlink" title="分治思想"></a>分治思想</h4><h5 id="介绍"><a href="#介绍" class="headerlink" title="介绍"></a>介绍</h5><p>把一个复杂的问题分成两个或更多的相同或相似的子问题，再把子问题分成更小的问题…直到最后子问题可以简单的直接求解，原问题的解即子问题的解的合并。</p>
<h5 id="基本思想和策略"><a href="#基本思想和策略" class="headerlink" title="基本思想和策略"></a>基本思想和策略</h5><p>分治法的设计思想是：<span style="color:red;">将一个难以直接解决的大问题，分割成一些规模较小的相同问题，以便各个击破，分而治之。</span></p>
<p>分治策略是：<span style="color:red;">对于一个规模为<code>n</code>的问题，将其分解成<code>k</code>个规模较小的子问题，这些子问题互相独立且与原问题形式相同，递归的解决这些子问题，然后将各子问题的解合并得到原问题的解。</span></p>
<h5 id="分治法的基本步骤"><a href="#分治法的基本步骤" class="headerlink" title="分治法的基本步骤"></a>分治法的基本步骤</h5><ul>
<li>分解：将原问题分解为若干个规模较小，互相独立，与原问题形式相同的子问题；</li>
<li>解决：若子问题规模较小而容易被解决则直接解决，否则递归的解各个子问题；</li>
<li>合并：将各个子问题的解合并为原问题的解。</li>
</ul>
<h5 id="代码实现demo"><a href="#代码实现demo" class="headerlink" title="代码实现demo"></a>代码实现demo</h5><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">package</span> org.example.soft.examination.data.structure.algorithm.thought;</span><br><span class="line"></span><br><span class="line"><span class="keyword">import</span> java.util.ArrayList;</span><br><span class="line"><span class="keyword">import</span> java.util.List;</span><br><span class="line"></span><br><span class="line"><span class="comment">/**</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@author</span> william</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@date</span> 2021/3/3 10:52</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"><span class="keyword">public</span> <span class="keyword">class</span> <span class="title class_">DivideConquer</span> {</span><br><span class="line"></span><br><span class="line">    <span class="keyword">public</span> <span class="keyword">static</span> List&lt;Integer&gt; <span class="title function_">diffWaysToCompute</span><span class="params">(String input)</span> {</span><br><span class="line">        List&lt;Integer&gt; ways = <span class="keyword">new</span> <span class="title class_">ArrayList</span>&lt;&gt;();</span><br><span class="line">        <span class="keyword">for</span> (<span class="type">int</span> <span class="variable">i</span> <span class="operator">=</span> <span class="number">0</span>; i &lt; input.length(); i++) {</span><br><span class="line">            <span class="type">char</span> <span class="variable">c</span> <span class="operator">=</span> input.charAt(i);</span><br><span class="line">            <span class="keyword">if</span> (c == <span class="string">'-'</span> || c == <span class="string">'+'</span> || c == <span class="string">'*'</span>) {</span><br><span class="line">                List&lt;Integer&gt; left = diffWaysToCompute(input.substring(<span class="number">0</span>, i));</span><br><span class="line">                List&lt;Integer&gt; right = diffWaysToCompute(input.substring(i + <span class="number">1</span>));</span><br><span class="line">                <span class="keyword">for</span> (<span class="type">int</span> l: left) {</span><br><span class="line">                    <span class="keyword">for</span> (<span class="type">int</span> r: right) {</span><br><span class="line">                        <span class="keyword">switch</span> (c) {</span><br><span class="line">                            <span class="keyword">case</span> <span class="string">'-'</span>:</span><br><span class="line">                                ways.add(l - r);</span><br><span class="line">                                <span class="keyword">break</span>;</span><br><span class="line">                            <span class="keyword">case</span> <span class="string">'+'</span>:</span><br><span class="line">                                ways.add(l + r);</span><br><span class="line">                                <span class="keyword">break</span>;</span><br><span class="line">                            <span class="keyword">case</span> <span class="string">'*'</span>:</span><br><span class="line">                                ways.add(l * r);</span><br><span class="line">                                <span class="keyword">break</span>;</span><br><span class="line">                            <span class="keyword">default</span>:</span><br><span class="line">                        }</span><br><span class="line">                    }</span><br><span class="line">                }</span><br><span class="line">            }</span><br><span class="line">        }</span><br><span class="line">        <span class="keyword">if</span> (ways.size() == <span class="number">0</span>) {</span><br><span class="line">            ways.add(Integer.valueOf(input));</span><br><span class="line">        }</span><br><span class="line">        <span class="keyword">return</span> ways;</span><br><span class="line">    }</span><br><span class="line"></span><br><span class="line">    <span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title function_">main</span><span class="params">(String[] args)</span> {</span><br><span class="line">        <span class="type">String</span> <span class="variable">input</span> <span class="operator">=</span> <span class="string">"2-11+99*12"</span>;</span><br><span class="line">        List&lt;Integer&gt; ans = diffWaysToCompute(input);</span><br><span class="line">        System.out.println(<span class="string">"ans = "</span> + ans);</span><br><span class="line">    }</span><br><span class="line">}</span><br><span class="line"></span><br></pre></td></tr></table></figure>
<h4 id="动态规划算法"><a href="#动态规划算法" class="headerlink" title="动态规划算法"></a>动态规划算法</h4><h5 id="介绍-1"><a href="#介绍-1" class="headerlink" title="介绍"></a>介绍</h5><p>它通常用于求解具有某种最优性质的问题。在这类问题中，可能会有许多可行解。每一个解都对应一个值，我们希望找到具有最优值的解。</p>
<p>动态规划算法和分治法类似，其基本思想也是将待求解的问题分解成若干个子问题，先求解子问题，然后从这些子问题的解得到原问题的解。</p>
<h5 id="分治和动态规划的区别"><a href="#分治和动态规划的区别" class="headerlink" title="分治和动态规划的区别"></a>分治和动态规划的区别</h5><ul>
<li>分治法往往用到递归计算，自上而下计算；而动态规划则直接自底向上计算；</li>
<li>分治法的小问题在递归的过程中可能会被反复计算，动态规划中的小问题计算后被存储，下次碰到时直接调用；</li>
<li>分治法的小问题的解只使用一次，动态规划的小问题的解可以作为上一级最优解的基础。</li>
</ul>
<h5 id="代码实现demo-1"><a href="#代码实现demo-1" class="headerlink" title="代码实现demo"></a>代码实现demo</h5><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">package</span> org.example.soft.examination.data.structure.algorithm.thought;</span><br><span class="line"></span><br><span class="line"><span class="keyword">import</span> org.apache.commons.lang3.time.StopWatch;</span><br><span class="line"></span><br><span class="line"><span class="comment">/**</span></span><br><span class="line"><span class="comment"> * 动态规划思想</span></span><br><span class="line"><span class="comment"> *</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@author</span> william</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@date</span> 2021/3/3 14:17</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"><span class="keyword">public</span> <span class="keyword">class</span> <span class="title class_">DynamicProgramming</span> {</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">    <span class="comment">/**</span></span><br><span class="line"><span class="comment">     * 爬楼梯</span></span><br><span class="line"><span class="comment">     * <span class="doctag">@date</span> 2021-03-03 14:19:56</span></span><br><span class="line"><span class="comment">     *</span></span><br><span class="line"><span class="comment">     * <span class="doctag">@param</span> n 楼梯的长度</span></span><br><span class="line"><span class="comment">     * <span class="doctag">@return</span> 多少种上楼梯的方法</span></span><br><span class="line"><span class="comment">     */</span></span><br><span class="line">    <span class="keyword">public</span> <span class="keyword">static</span> <span class="type">int</span> <span class="title function_">climbStairs</span><span class="params">(<span class="type">int</span> n)</span> {</span><br><span class="line">        <span class="comment">// 分治的思想</span></span><br><span class="line">        <span class="keyword">if</span> (n &lt;= <span class="number">2</span>) {</span><br><span class="line">            <span class="keyword">return</span> n;</span><br><span class="line">        }</span><br><span class="line">        <span class="keyword">return</span> climbStairs(n - <span class="number">1</span>) + climbStairs(n - <span class="number">2</span>);</span><br><span class="line">    }</span><br><span class="line"></span><br><span class="line">    <span class="keyword">public</span> <span class="keyword">static</span> <span class="type">int</span> <span class="title function_">climbStairsTwo</span><span class="params">(<span class="type">int</span> n)</span> {</span><br><span class="line">        <span class="comment">// 动态规划的思想</span></span><br><span class="line">        <span class="keyword">if</span> (n &lt;= <span class="number">2</span>) {</span><br><span class="line">            <span class="keyword">return</span> n;</span><br><span class="line">        }</span><br><span class="line">        <span class="type">int</span> <span class="variable">pre1</span> <span class="operator">=</span> <span class="number">2</span>, pre2 = <span class="number">1</span>;</span><br><span class="line">        <span class="keyword">for</span> (<span class="type">int</span> <span class="variable">i</span> <span class="operator">=</span> <span class="number">2</span>; i &lt; n; i++) {</span><br><span class="line">            <span class="type">int</span> <span class="variable">curr</span> <span class="operator">=</span> pre1 + pre2;</span><br><span class="line">            pre2 = pre1;</span><br><span class="line">            pre1 = curr;</span><br><span class="line">        }</span><br><span class="line">        <span class="keyword">return</span> pre1;</span><br><span class="line">    }</span><br><span class="line"></span><br><span class="line">    <span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title function_">main</span><span class="params">(String[] args)</span> {</span><br><span class="line">        <span class="type">int</span> <span class="variable">n</span> <span class="operator">=</span> <span class="number">44</span>;</span><br><span class="line">        <span class="type">StopWatch</span> <span class="variable">watch</span> <span class="operator">=</span> <span class="keyword">new</span> <span class="title class_">StopWatch</span>();</span><br><span class="line">        watch.start();</span><br><span class="line">        <span class="type">int</span> <span class="variable">climbAns</span> <span class="operator">=</span> climbStairs(n);</span><br><span class="line">        watch.stop();</span><br><span class="line">        System.out.println(<span class="string">"climbStairs耗时："</span> + watch.getNanoTime());</span><br><span class="line">        watch.reset();</span><br><span class="line">        watch.start();</span><br><span class="line">        <span class="type">int</span> <span class="variable">climbTwoAns</span> <span class="operator">=</span> climbStairs(n);</span><br><span class="line">        watch.stop();</span><br><span class="line">        System.out.println(<span class="string">"climbStairsTwo耗时："</span> + watch.getNanoTime());</span><br><span class="line">        System.out.println(<span class="string">"climbAns = "</span> + climbAns);</span><br><span class="line">        System.out.println(<span class="string">"climbTwoAns = "</span> + climbTwoAns);</span><br><span class="line">    }</span><br><span class="line">}</span><br><span class="line"></span><br></pre></td></tr></table></figure>
<h5 id="其他的一些动态规划的题目"><a href="#其他的一些动态规划的题目" class="headerlink" title="其他的一些动态规划的题目"></a>其他的一些动态规划的题目</h5><ul>
<li>强盗抢劫</li>
<li>强盗在环形街抢劫</li>
<li>信件错排</li>
<li>母牛生产</li>
<li>矩阵路径<ul>
<li>矩阵的最小路径和</li>
<li>矩阵的总路径数</li>
</ul>
</li>
<li>数组区间<ul>
<li>数组区间和</li>
<li>子数组最大的和</li>
<li>数组中等差递增子区间的个数</li>
</ul>
</li>
<li>分割整数<ul>
<li>分割整数的最大乘积</li>
<li>按平方数来分割整数</li>
<li>分割整数构成字母字符串</li>
</ul>
</li>
<li>最长递增子序列<ul>
<li>最长递增子序列</li>
<li>一组整数能够构成的最长链</li>
<li>最长摆动子序列</li>
</ul>
</li>
<li>最长公共子序列</li>
<li>背包<ul>
<li>划分数组为相等的两部分</li>
<li>改变一组数的正负号使得它们的和为一给定数</li>
<li>字符串按单词列表分割</li>
<li>字符构成最多的字符串</li>
<li>找零钱的最小硬币数</li>
<li>组合总和</li>
</ul>
</li>
<li>股票交易<ul>
<li>需要冷却期的股票交易</li>
<li>需要交易费用的股票交易</li>
<li>买入和售出股票最大的收益</li>
<li>只能进行两次的股票交易</li>
<li>只能进行K次的股票交易</li>
</ul>
</li>
<li>字符串编辑<ul>
<li>删除两个字符串的字符使它们相等</li>
<li>编辑距离</li>
<li>复制粘贴字符</li>
</ul>
</li>
</ul>

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